We study the problem of delegated choice with inspection cost (DCIC), which is a variant of the delegated choice problem by Kleinberg and Kleinberg (EC'18) as well as an extension of the Pandora's box problem with nonobligatory inspection (PNOI) by Doval (JET'18). In our model, an agent may strategically misreport the proposed element's utility, unlike the standard delegated choice problem which assumes that the agent truthfully reports the utility for the proposed alternative. Thus, the principal needs to inspect the proposed element possibly along with other alternatives to maximize its own utility, given an exogenous cost of inspecting each element. Further, the delegation itself incurs a fixed cost, thus the principal can decide whether to delegate or not and inspect by herself. We show that DCIC indeed is a generalization of PNOI where the side information from a strategic agent is available at certain cost, implying its NP-hardness by Fu, Li, and Liu (STOC'23). We first consider a costless delegation setting in which the cost of delegation is free. We prove that the maximal mechanism over the pure delegation with a single inspection and an PNOI policy without delegation achieves a $3$-approximation for DCIC with costless delegation, which is further proven to be tight. These results hold even when the cost comes from an arbitrary monotone set function, and can be improved to a $2$-approximation if the cost of inspection is the same for every element. We extend these techniques by presenting a constant factor approximate mechanism for the general setting for rich class of instances.

翻译：本研究探讨代价性审查委托选择问题，该问题是Kleinberg与Kleinberg（EC'18）所提出委托选择问题的变体，同时也是Doval（JET'18）提出的非强制性审查潘多拉盒问题的扩展。在我们的模型中，代理方可能策略性地误报所提议元素的效用值，这与标准委托选择问题中假设代理方如实报告提议方案效用的设定不同。因此，委托方在给定外生性元素审查成本的前提下，可能需要审查提议元素及其他备选方案以实现自身效用最大化。此外，委托行为本身会产生固定成本，故委托方可自行决定是否采取委托或自主审查。我们证明代价性审查委托选择问题实质上是非强制性审查潘多拉盒问题的广义形式，其中策略性代理方提供的侧向信息需付出特定成本获取，这通过Fu、Li与Liu（STOC'23）的研究暗示了该问题的NP难解性。我们首先考察无成本委托场景，其中委托行为不产生成本。我们证明：在无成本委托的代价性审查委托选择问题中，基于单次审查的纯委托机制与无委托的非强制性审查潘多拉盒策略的最大化机制可实现3倍近似比，该结果被进一步证明是紧致的。这些结论在成本来源于任意单调集函数时依然成立，且当各元素审查成本相同时可提升至2倍近似比。通过扩展这些技术，我们为广泛实例类别的一般性场景提出了具有常数倍近似比的机制。